Cpu/mobo possibly dead

so I have with all i have currently which is limited to narrowed it down to either dead cpu or dead mobo. I do not have any spare parts green light is still lit on the mother board. when I push the power button fans spur up for a second then nothing. the only issue thru my 3 years of my pc is heating issues with the cpu aka overwatch per say the past year at which I kept my room very cool as well as some fans blowing towards the pc. pc suddendly turned off and would not turn off during use and was not doing any gaming was just downloading path of exile and then it suddenly shutdown.

Will a motherboard speaker help me determine my problem

I have limited funds and only the possibility to replace the mobo or cpu as well as get addition cooling system or better for the cpu

motherboard is ASUS M5A97 R2.0 AM3+ AMD 970 cpu is AMD FX-8350 Black Edition Vishera 8-Core 4.0 GHz